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In this chapter, we will look at using the standard deviation as a measuring stick and some properties of data sets that are normally distributed. The number of standard deviations a particular observation in a data set is away from the mean of that set is called it , and is denoted zx. It is calculated as follows: zx = x-x x -m or zx = s s If zx > 0, then the observation is above average If zx < 0, then the observation is below average z-scores have no units; when observations are changed into their z-scores it is called Once standardized, observations from different data sets having different centers and spreads can be compared. In the year 2009, cars sold in the United States had an average of 135 horsepower with a standard deviation of 40 horsepower. (a) Find the z-score for a car with 120 horsepower. (b) Find the z-score for a car with 200 horsepower. (c) A certain car has z-score of 1.2. How much horsepower does it have? (d) A certain car has z-score of 0.6. How much horsepower does it have? Adult Dalmatians have an average weight of 52 pounds with a standard deviation of 6 pounds. Adult German Shepherds have an average weight of 77 pounds with a standard deviation of 3.6 pounds. Which is more unusual, an adult Dalmatian weighing 63 pounds or an adult German Shepherd weighing only 68 pounds? If the histogram of a data set is unimodal and fairly symmetric, then its distribution is called l. If such a distribution has mean m and standard deviation , it can be modeled with a normal model, the notation of which , and the shape is below: N ( m, sis) - 3 - 2 - + + 2 + 3 The standard normal curve is N ( 0, 1) . This is commonly called the z–curve. It has shape below: -3 -2 -1 0 1 2 3 For a normal distribution: ≈ 68% of values fall within one standard deviation of the mean ≈ 95% of values fall within two standard deviations of the mean ≈ 99.7% of values fall within three standard deviations of the mean